Determine the variances and covariance

Question

Solve the following problem:

Consider a two-user, synchronous CDMA transmission system, where the received signal is

r(t) = √ε₁b₁g₁(t) + √ε₂b₂g₂(t) + n(t),        0 ≤ t ≤ T

and (b₁, b₂) = (±1, ±1). The noise process n(t) is zero-mean Gaussian and white, with spectral density N₀/2. The demodulator for r(t) is shown in Figure .

a. Show that the correlator outputs r₁ and r₂ at t = T may be expressed as

r₁ = √ε₁b₁ + √ε₂pb₂ + n₁

r₂ = √ε₁b₁p + √ε₂b₂ + n₂

b. Determine the variances of n1 and n2 and the covariance of n₁ and n₂.

c. Determine the joint PDF p(r₁,r₂|b₁, b₂).